Chapter 7: Q. 8 (page 631)

If you suspect that a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges, explain why you would need to compare the series with a divergent series, using either the comparison test or the limit comparison test.

Short Answer

Expert verified

As a result, if the ${鈭�}_{k = 1}^{鈭�} a_{k}$ series diverges, it must be compared to a divergent series.

Step by step solution

Step 1. Given information

A series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is given in the question

Step 2. Verification

The limit comparison test for ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ are the series having positive terms the the following conditions may apply,

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , L must be positive number then it may be either converging or diverging.

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ then if ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ then if ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges then ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges

If the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is divergent, comparing it to convergent series will yield no results since the behaviour of the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is dependent on the behaviour of the series ${鈭�}_{k = 1}^{鈭�} b_{k}$ .

As a result, if the ${鈭�}_{k = 1}^{鈭�} a_{k}$ series diverges, it must be compared to a divergent series.

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If you suspect that a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges, explain why you would need to compare the series with a divergent series, using either the comparison test or the limit comparison test.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Verification

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91影视

If you suspect that a series鈭�k=1鈭�ak diverges, explain why you would need to compare the series with a divergent series, using either the comparison test or the limit comparison test.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Discrete Mathematics

Decision Maths

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

If you suspect that a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges, explain why you would need to compare the series with a divergent series, using either the comparison test or the limit comparison test.